The course intends to provide the students the fundaments of the modern theory of partial differential equations (PDE). In the first part the notion of function space will be studied: the L^p spaces of Lebesgue, Banach spaces, Hilbert spaces. In the second part it is shown that these spaces are the natural environment in which one obtains existence and uniqueness theorems for a large class of PDE's.
Expected learning outcomes
At the end of the course the student will have acquired the following capabilities: he/she will 1) know the structure and properties of the Lebesgue spaces L^p 2) have a good knowledge of the properties of Banach and Hilbert spaces 3) know the properties of Sobolev spaces 4) have seen the fundamental theorems of compactness: the theorems of Ascoli-Arzela and of Rellich-Kondrachov 5) know the weak formulation of elliptic equations of second order 6) will know to apply the Dirichlet principle to linear and nonlinear elliptic equations 7) will know the importance of the regularity theorems for weak solutions, and will have seen the corresponding theorems 8) will have studied the heat eqqation and the representation formulas of the solutions 9) will know the modern theory of second order parabolic equations, weak solutions and energy estimates 10) will have studied the basics of hyperbolic equations
Lesson period: Second semester
(In case of multiple editions, please check the period, as it may vary)
· L^p spaces · Banach spaces and Hilbert spaces · Introduction to distributions · Sobolev spaces · Weak formulation of second order elliptic equations · The Dirichlet principle · Regularity of weak solutions · Parabolic equations of second order: weak solutions and regularity
Prerequisites for admission
Mathematical analysis 3
Lectures in traditional mode, at the blackboard.
L.E. Evans, Partial Differential Equations, AMS, (1998) H. Brezis, Analyse fonctionnelle, Masson, (1983) D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer, (1977) E. Lieb, M. Loss, Analysis, GTM in Mathematics, vol. 14, AMS, (1997)
Assessment methods and Criteria
The exam consists of a single oral exam (30 minutes) which serves to verify the theoretical knowledge acquired during the course and the ability to solve exercises similar to the one which were proposed during the course.